Tensors for Physics

5.2 Principal Values 57

the tensorSisisotropic, i.e. it is proportional to the unit tensor:

Sμν=Sδμν. (5.6)

This follows from (5.4), due to the completeness relation for the orthogonal unit
vectorse(i), viz.,
∑^3

i= 1

eμ(i)eν(i)=δμν. (5.7)

By definition, an isotropic tensor has noanisotropicpart. Whenever the three
eigenvalues are not equal to each other, the symmetric tensor possesses a non-zero
traceless part, sometimes also calledanisotropic part. Two cases can be distin-
guished: (i) two eigenvalues are equal, but different from the third one, and (ii)
three different eigenvalues. These cases are referred to with the labelsuniaxialand
biaxial.

5.2.3 Uniaxial Tensors.

When only two the principal values are equal but different from the third one, say
S(^1 )=S(^2 )=S(^3 ), the tensor is calleduniaxial. It possesses a symmetry axis which
is parallel toe(^3 )in the special case considered. It is convenient to use the notation
e(^3 )=efor the unit vector parallel to the symmetry axis and to denote the eigenvalues
associated with the directions parallel and perpendicular to this direction byS‖and
S⊥, respectively. This means:

S(^1 )=S(^2 )=S⊥, S(^3 )=S‖.

Thus the uniaxial tensor can be written as

Sμν=S‖eμeν+S⊥(δμν−eμeν), (5.8)

and

Sμν=

1

3

(S‖+ 2 S⊥)δμν+(S‖−S⊥)eμeν. (5.9)

In matrix notation, this expression is equivalent to

S:=

1

3

(S‖+ 2 S⊥)

⎛

⎝

100

010

001

⎞

⎠+^2

3

(S‖−S⊥)

⎛

⎝

−^1200

0 −^120

001

⎞

⎠. (5.10)

The factor^23 in the symmetric traceless part stems fromeμeνeμeν =^23.